= Integrating amplifier
An \b[integrating amplifier] produces an output proportional to the time integral of its input over an intended operating range. A common circuit uses an <operational amplifier>, an input <resistor> $R$ and a feedback <capacitor> $C$. With the non-inverting input held at zero, ideal negative feedback maintains a <virtual ground> at the inverting input. Negligible amplifier input current then gives
$$
\frac{v_{\rm in}}R=-C\frac{dv_{\rm out}}{dt},\qquad
v_{\rm out}(t)=v_{\rm out}(t_0)-\frac1{RC}\int_{t_0}^{t}v_{\rm in}(s)\,ds.
$$
The initial output records the capacitor's initial charge. Texas Instruments' circuit note derives this inverting relation and explains why real integration is limited by the circuit time constant and amplifier <bandwidth>.<Integrator Circuit by Pete Semig>[pp1–2, Design Description and ideal equation]{ref}
For an original example, choose $R=100\,\mathrm{k\Omega}$ and $C=10\,\mathrm{nF}$, so $RC=1\,\mathrm{ms}$. A constant0.20volt input lasting2milliseconds changes the output by $-0.40\,\mathrm V$. Repeating ten such pulses without intervening input or <integrator reset> changes it by $-4.0\,\mathrm V$ in the ideal model. The output retains information about accumulated pulse area, rather than only the present input value.
A small offset also accumulates. With the same components, a residual input-equivalent1millivolt gives a slope of $-1\,\mathrm{V\,s^{-1}}$. Starting from zero, an idealized negative output limit at5volts would be reached in5seconds. The finite supply range causes <amplifier clipping>; an ideal integral extending indefinitely is not physically available.
Putting a resistor $R_f$ in parallel with the feedback capacitor supplies a DC path. Under ideal amplifier assumptions its zero-state <transfer function> is
$$
H(s)=-\frac{R_f/R}{1+sR_fC}.
$$
At frequencies where $|sR_fC|\gg1$, this approaches $-1/(sRC)$; at low frequency it instead approaches the finite gain $-R_f/R$. The resistor prevents unrestricted DC accumulation while changing the lowest usable integration frequency.<Integrator Circuit by Pete Semig>[p1, feedback-resistor purpose and Design Notes]{ref}
An <integrating amplifier> can therefore turn a short detector pulse into a measurable accumulated output, but its initial condition, reset, baseline, frequency range and output limits belong to the measurement model. A <charge amplifier> is a related current-input arrangement in which collected charge is converted through feedback capacitance.
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